A387263 Partial sums of A247287.

0, 0, 1, 5, 18, 56, 164, 468, 1325, 3751, 10653, 30381, 87003, 250095, 721300, 2086308, 6049629, 17580415, 51187605, 149293221, 436088521, 1275559185, 3735597612, 10952250100, 32143070778, 94422309606, 277608161509, 816829091513, 2405170198470, 7086849120836

Offset

0,4

Comments

Second partial sums of A097861.

For p Pythagorean prime (A002144), a(p) - 1 == 0 (mod p).

For p prime of the form 4*k + 3 (A002145), a(p) + 1 == 0 (mod p).

Links

Paolo Xausa, Table of n, a(n) for n = 0..2000

Formula

G.f.: ((1-x)/sqrt(1-2*x-3*x^2)-1) / (2*(1-x)^3).

E.g.f.: exp(x)*(g(x) + 2*Integral_{x=-oo..oo} g(x) dx + Integral_{x=-oo..oo} (Integral_{x=-oo..oo} g(x) dx) dx) where g(x) = (-1+BesselI(0, 2*x))/2.

D-finite with recurrence n*a(n) = (6*n-3)*a(n-1) - (11*n-11)*a(n-2) + (4*n-6)*a(n-3) + (9*n-18)*a(n-4) - (10*n-25)*a(n-5) + (3*n-9)*a(n-6).

a(n) = Sum_{j=0..n}(Sum_{k=0..j} A390369(j, k)).

a(n) = a(n-2) - (n+1) + A383527(n) for n > 1.

a(n) = (A385641(n) - binomial(n+2, 2)) / 2.

From Mélika Tebni, Dec 29 2025: (Start)

a(n) = Sum_{k=0..n} A391467(n, k).

a(n) = (Sum_{k=0..n} ((n-k+1)*A002426(k) - k) - (n+1)) / 2.

For n >= 2, 2*a(n) - 4*a(n-1) + 2*a(n-2) = A180282(n) = A002426(n) - 1. (End)

Maple

a := series(exp(x)*((-1+BesselI(0, 2*x))/2 + 2*int((-1+BesselI(0, 2*x))/2, x) + int(int((-1+BesselI(0, 2*x))/2, x), x)), x = 0, 30):
seq(n!*coeff(a, x, n), n = 0 .. 29);
# Recurrence:
a := proc (n) option remember; `if`(n < 7, [0, 0, 1, 5, 18, 56, 164][n+1], 1/n*((6*n-3)*a(n-1) - (11*n-11)*a(n-2) + (4*n-6)*a(n-3) + (9*n-18)*a(n-4) - (10*n-25)*a(n-5) + (3*n-9)*a(n-6))) end: seq(a(n), n = 0 .. 29);

Mathematica

Module[{x}, CoefficientList[Series[((1 - x)/Sqrt[1 - 2*x - 3*x^2] - 1)/(2*(1 - x)^3), {x, 0, 30}], x]] (* Paolo Xausa, Dec 29 2025 *)

Prog

(Python)
from math import comb
def a(n):
    return (sum(comb(n+1, k+1)*comb(2*(k//2), k//2) for k in range(n + 1)) - comb(n+2, 2))//2
print([a(n) for n in range(30)])

Crossrefs

Cf. A002144, A002145, A002426, A097861, A180282, A247287, A383527, A385641, A390369, A391467.

Sequence in context: A317849 A307572 A325923 * A335720 A093374 A258109

Adjacent sequences: A387260 A387261 A387262 * A387264 A387265 A387266

Keyword

nonn,easy

Author

Mélika Tebni, Nov 25 2025

Status

approved